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XXXVII. On the Approximate Drawing of Circular Arcs of given 

 Lengths. By W. J. Macquorn Rankine, C.E., LL.D., 

 F.R.8.* 



1 . THf ULES. — The following rules for approximately drawing 

 ■*■*■* circular arcs of given lengths have not, so far as I know, 

 been hitherto published. 



Rule I, (See fig. 1.) — From a Fig. 1. 



given point A, in the circumfe- 

 rence of a given circle A B, to lay 

 off an arc which shall be approxi- 

 mately of a given length. From 

 A draw the straight tangent A C, 

 equal to one-fourth part of the given 

 length. About C, with a radius 

 equal to three-fourths of the given 

 length, draw a circular arc cutting 

 the given circle in D. Then AD 

 will be the arc required. 



Rule II. (See fig. 1.) — To draw a circular arc which shall touch 

 a given straight line AC at a given point A, shall subtend a 

 given angle, and shall be approximately of a given length. Make 

 A C equal to one-fourth of the given length ; about C, with a 

 radius equal to three-fourths of the given length, describe a 

 circle ; draw the straight line A D, making the angle CAD 

 equal to one-half of the given angle, and cutting the circle in 

 D ; D will be the other end of the required arc. Bisect AD in 

 E; draw the straight lines A F perpendicular to A C, and E F 

 perpendicular to A. D, meeting each other in F ; that point will 

 be the centre for the required arc A D. 



2. Extent of Errors. — The arcs laid off according to the pre- 

 ceding rules are somewhat longer than the exact length given. 

 The following are examples of the proportionate errors : — 



Error in fractions 

 of the arc, about 

 In laying off an arc subtending 30° .... t^Joo* 



» » » 45° .... g 900* 



71 » „ 60 ... . 9J5. 



The proportionate error varies nearly as the fourth power of the 

 angle subtended by the arc, and diminishes very rapidly when 

 that angle becomes small. 



3. Demonstration. — In fig. 2, let a straight line A G and an 



* Communicated by the Author, having been read at the British Asso- 

 ciation, Dundee Meeting, September ] 86/. 



